Phase space is a mathematical framework used in physics and dynamical systems to represent all possible states of a system. Each point in phase space corresponds to a unique state of the system, defined by its position and momentum (or other relevant variables like velocity, energy, etc.).
For a system with nn degrees of freedom, the phase space is 2n-dimensional, consisting of:
- n position coordinates, and
- n momentum (or velocity) coordinates.
Key Features:
- The trajectory of a system through phase space shows how its state evolves over time.
- Fixed points (equilibria), cycles, and attractors are represented as distinct geometrical features in phase space.
- Used to analyze both regular (periodic) and chaotic behavior.
Examples:
- A simple pendulum has a 2D phase space with angle and angular velocity.
- In classical mechanics, the motion of planets is described in a high-dimensional phase space.
- In thermodynamics, systems are analyzed using generalized coordinates (volume, pressure, energy).
Phase space provides a complete visual and analytical tool for studying dynamical systems, helping scientists understand motion, stability, and long-term behavior in both classical and quantum contexts.